Following the "standard" for conodont reconstructions, morphology is based on the 3 species with known soft tissues (Clydagnathus windsorensis, Panderodus unicostatus, and the giant Promissum pulchrum) (Aldridge et al. 1986, Gabbott et al. 1995, Murdock & Smith 2021), with details filled in from living hagfish and lampreys based on the assumed vertebrate (and possibly even cyclostome) affinity of conodonts (Miyashita et al. 2019). The count of 7 pairs of gill openings (as in lampreys) is simply because i couldn't be bothered to sculpt more.

Note that the mouth is not depicted as a the usual gaping hole filled with spiny elements but rather as folded tissues nicely hiding any trace of offensive toothiness, much like modern hagfish, which, despite their impressive set of "teeth", have a very kissable (closed) mouth. I understand the didactic value of showing the element apparatus in conodont reconstructions but have always felt a little weird about depicting animals actually swimming around looking like that... but who knows?

Another departure from the usual way of reconstructing conodonts is the inclusion of a single nostril. This is based on the single nostril of extant hagfish and lamprey (to which (eu-)conodonts may be most closely related to) (Miyashita et al. 2019), and also supported by the fact that a single nostril may be part of the ancestral state of vertebrates (Oisi et al. 2013) (assuming conodonts are actually vertebrates, of course).

Anyway, that was a lot of reading and shoddy speculation for a background model. Certainly don't trust any of it, I don't know shit about conodonts.

References:

Aldridge, R. J., Briggs, D. E. G., Clarkson, E. N. K., & Smith, M. P. (1986). The affinities of conodonts—New evidence from the Carboniferous of Edinburgh, Scotland. Lethaia, 19(4), 279–291. https://doi.org/10.1111/j.1502-3931.1986.tb00741.x

Gabbott, S. E., Aldridge, R. J., & Theron, J. N. (1995). A giant conodont with preserved muscle tissue from the Upper Ordovician of South Africa. Nature, 374(6525), 800–803. https://doi.org/10.1038/374800a0

Miyashita, T., Coates, M. I., Farrar, R., Larson, P., Manning, P. L., Wogelius, R. A., Edwards, N. P., Anné, J., Bergmann, U., Palmer, A. R., & Currie, P. J. (2019). Hagfish from the Cretaceous Tethys Sea and a reconciliation of the morphological–molecular conflict in early vertebrate phylogeny. Proceedings of the National Academy of Sciences of the United States of America, 116(6), 2146–2151. https://doi.org/10.1073/pnas.1814794116

Murdock, D. J. E., & Smith, M. P. (2021). Panderodus from the Waukesha Lagerstätte of Wisconsin, USA: A primitive macrophagous vertebrate predator. Papers in Palaeontology, 7(4), 1977–1993. https://doi.org/10.1002/spp2.1389